Letters of the word $\bf{"ALASKA"}$ can be arranged in a circle (Diff. b/w clockwise & Anticlockwise.) (1): Numbers of ways in which all the letters of the word $\bf{"ALASKA"}$ can be arranged in a circle distinguishing between the clockwise and anticlockwise arrangements, is
(2): Numbers of ways in which all the letters of the word $\bf{"SUNDAY"}$ can be arranged in a circle distinguishing between clockwise and anticlockwise arrangements, is
$\bf{My\; Try::}$ For $\bf{1^{st}}$ one:
If we arrange the letters of the word $\bf{"ALASKA"}$ in a row, then the total no. of ways $\displaystyle = \frac{6!}{3!} = 120$
But I did not understand how can I arrange these words along a circle.
Can anyone give me a detailed explanation?
Thanks. 
 A: The six distinct letters of the word SUNDAY can be arranged in $6!$ ways along a line.  However, if they are arranged along a circle, the six permutations SUNDAY, UNDAYS, NDAYSU, DAYSUN, AYSUND, YSUNDA obtained by successive $60^\circ$ rotations are indistinguishable.  
More generally, the six permutations that can be obtained by rotations of $0^\circ$, $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, and $300^\circ$ are indistinguishable.  Hence, there are 
$$\frac{6!}{6} = 5!$$ 
indistinguishable clockwise permutations of the word SUNDAY.  By the same reasoning, there are $5!$ anti-clockwise permutations of the word SUNDAY, each of which corresponds to reading a given clockwise permutation in reverse order.  Since we are distinguishing between clockwise and anti-clockwise arrangements, we count a given permutation and its reflection separately.  Therefore, there are $5!$ circular permutations of the word SUNDAY.
As you observed, there are $$\frac{6!}{3!} = 120$$ distinct arrangements of the six letters of the word ALASKA in a line.  Since there are six indistinguishable arrangements produced by rotations around the circle, there are $$\frac{6!}{6 \cdot 3!} = 20$$ distinct clockwise arrangements of the letters of the word ALASKA.  Since we are treating reflections as distinguishable, there are $20$ distinguishable circular permutations of the word ALASKA.
A: Hint: Supppose you have an $n$-letter word written on a circle. In how many ways can you read it (sequentially, clockwise)?
A: (1) Distinguish one of the positions in the circle and place the letter L in it.  The remaining letters can then be arranged in 
$$\frac{5!}{3!} = 20$$ ways.
